Abstract. Let Ap denote the class of holomorphic functions on the unit disc whose first ^-derivatives belong to the disc algebra. We characterize the boundary interpolation sets for Ap, that is, those closed sets E c T such that every function in C(E) extends to a function in Ap.We also give a constructive proof of the corresponding result for A °° (see [1]). We show that the structure of these sets is in some sense related to BMO and that this fact can be used to obtain precise estimates of outer functions vanishing on E.